Aout the quotient topology

Question

Let $X$ be a topological space, $\mathcal{R}$ an equivalence relation on $X$ and $p$ the projection map.

If $X$ was a separable space, can we said that the quotient space $X/\mathcal{R}$ is separable?

Thank you all.

Answer 1

Let $\pi$ be the quotient map. Let $\{x_n\}$ be dense in $X$. If $U$ is a nonempty open set in the quotient then $\pi^{-1}(U)$ is a nonempty open set in X. Hence some $x_n$ belongs to $\pi^{-1}(U)$ and the equivalence class of $x_n$ belongs to $U$. This proves that equivalence classes of $x_n$'s form a countable dense set in $X/\mathcal R$.